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Leo Lam © Signals and Systems EE235
Leo Lam © Breeding What do you get when you cross an elephant and a zebra? Elephant zebra sin theta
Leo Lam © Today’s menu Friday: LCCDE Zero-Input Response Today: LCCDE Zero-State Response (forced)
Zero input response (example) Leo Lam © steps to solving Differential Equations: Step 1. Find the zero-input response = natural response y n (t) Step 2. Find the Particular Solution y p (t) Step 3. Combine the two Step 4. Determine the unknown constants using initial conditions
From Friday (example) Leo Lam © Solve Guess solution: Substitute: We found: Solution: Characteristic roots = natural frequencies/ eigenvalues Unknown constants: Need initial conditions
Zero-state output of LTI system Leo Lam © Response to our input x(t) LTI system: characterize the zero-state with h(t) Initial conditions are zero (characterizing zero-state output) Zero-state output: Total response(t)=Zero-input response (t)+Zero-state output(t) T (t) h(t)
Zero-state output of LTI system Leo Lam © Zero-input response: Zero-state output: Total response: Total response(t)=Zero-input response (t)+Zero-state output(t) “Zero-state”: (t) is an input only at t=0 Also called: Particular Solution (PS) or Forced Solution
Zero-state output of LTI system Leo Lam © Finding zero-state output (Particular Solution) Solve: Or: Guess and check Guess based on x(t)
Trial solutions for Particular Solutions Leo Lam © Guess based on x(t) Input signal for time t> 0 x(t) Guess for the particular function y P
Particular Solution (example) Leo Lam © Find the PS (All initial conditions = 0): Looking at the table: Guess: Its derivatives:
Particular Solution (example) Leo Lam © Substitute with its derivatives: Compare:
Particular Solution (example) Leo Lam © From We get: And so:
Particular Solution (example) Leo Lam © Note this PS does not satisfy the initial conditions! Not 0!
Natural Response (doing it backwards) Leo Lam © Guess: Characteristic equation: Therefore:
Complete solution (example) Leo Lam © We have Complete Sol n : Derivative:
Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Complete Sol n : Derivative: Subtituting: Two equations Two unknowns
Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Solving: Subtitute back: Then add u(t): y n ( t ) y p ( t ) y ( t )
Another example Leo Lam © Solve: Homogeneous equation for natural response: Characteristic Equation: Therefore: Input x(t)
Another example Leo Lam © Solve: Particular Solution for Table lookup: Subtituting: Solving: b=-1, =-2 No change in frequency! Input signal for time t> 0 x(t) Guess for the particular function y P
Another example Leo Lam © Solve: Total response: Solve for C with given initial condition y(0)=3 Tada!
Stability for LCCDE Leo Lam © Stable with all Re( j <0 Given: A negative means decaying exponentials Characteristic modes
Stability for LCCDE Leo Lam © Graphically Stable with all Re( j )<0 “Marginally Stable” if Re( j )=0 IAOW: BIBO Stable iff Re(eigenvalues)<0 Im Re Roots over here are stable
Leo Lam © Summary Differential equation as LTI system Stability of LCCDE
Leo Lam © Signals and Systems EE235 Lecture 19.
Leo Lam © Signals and Systems EE235 Leo Lam © Stanford The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake,
WEEK 1 You have 10 seconds to name…
Addition 1’s to
Addition Facts = = =
TWO STEP EQUATIONS 1. SOLVE FOR X 3. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE 2. DO THE ADDITION STEP FIRST.
Leo Lam © Signals and Systems EE235. Leo Lam © Today’s menu Yesterday: Exponentials Today: Linear, Constant-Coefficient Differential.
SUBTRACTING INTEGERS 1. CHANGE THE SUBTRACTION SIGN TO ADDITION 2. TAKE THE INVERSE OF THE SECOND NUMBER 3. FOLLOW THE RULES FOR ADDITION 4. ADD THE OPPOSITE.
25 seconds left….. 24 seconds left….. 23 seconds left…..
DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.
© S Haughton more than 3?
1 Unit 1 Kinematics Chapter 1 Day
1 Copyright © 2010, Elsevier Inc. All rights Reserved Fig 2.1 Chapter 2.
Leo Lam © Signals and Systems EE235 Leo Lam © Today’s menu Exponential response of LTI system LCCDE Midterm Tuesday next week.
Squares and Square Root WALK. Solve each problem REVIEW:
MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.
Jeopardy Topic 1Topic Q 1Q 6Q 11Q 16Q 21 Q 2Q 7Q 12Q 17Q 22 Q 3Q 8Q 13Q 18Q 23 Q 4Q 9Q 14Q 19Q 24 Q 5Q 10Q 15Q 20Q 25 Final Jeopardy.
Leo Lam © Signals and Systems EE235 Lecture 18.
Leo Lam © Signals and Systems EE235. Courtesy of Phillip Leo Lam ©
Solve an equation by multiplying by a reciprocal EXAMPLE 5 SOLUTION Write original equation. x = 4x = – x = – Solve 7 2 – The coefficient.
MULTIPLICATION EQUATIONS 1. SOLVE FOR X 3. WHAT EVER YOU DO TO ONE SIDE YOU HAVE TO DO TO THE OTHER 2. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE.
We will resume in: 25 Minutes We will resume in: 24 Minutes.
By D. Fisher Geometric Transformations. Reflection, Rotation, or Translation 1.
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ABC Technology Project Mrs. Kiddle. ABCs of Technology Word 1 Word 2 Word 3 Word 4 Word 5 Word 6 Word 7 Word 8 Word 9 Word 19 Word 20 Word 21 Word 22.
Copyright © Cengage Learning. All rights reserved. Quadratic Equations, Quadratic Functions, and Complex Numbers 9.
Business Transaction Management Software for Application Coordination 1 Business Processes and Coordination.
ADDING INTEGERS 1. POS. + POS. = POS. 2. NEG. + NEG. = NEG. 3. POS. + NEG. OR NEG. + POS. SUBTRACT TAKE SIGN OF BIGGER ABSOLUTE VALUE.
Leo Lam © Signals and Systems EE235. Leo Lam © Happy Tuesday! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).
Simultaneous Equations elimination. What are they? Simply 2 equations –With 2 unknowns –Usually x and y To SOLVE the equations means we find values of.
Chapter 13 How Cells Obtain Energy from Food Essential Cell Biology Third Edition Copyright © Garland Science 2010.
MULTIPLYING MONOMIALS TIMES POLYNOMIALS (DISTRIBUTIVE PROPERTY)
Figure 12-1 Essential Cell Biology (© Garland Science 2010)
Copyright © 2011, Elsevier Inc. All rights reserved. Chapter 5 Author: Julia Richards and R. Scott Hawley.
ALGEBRAIC EXPRESSIONS Step 1 Write down the question Step 2 Plug in the numbers Step 3 Use PEMDAS Work down, Show all steps.
Past Tense Probe Past Tense Probe – Practice 1 Past Tense Probe – Practice 2.
Substitution. Take OUT the variable and PUT in the new expression y = 4x + 1 2x – y = 9 y = 4x + 1 2x – y = 9 -2x – 3 = 6x + 1 y = -2x -3 y = 6x + 1 y.
Reading Assignment: Chapter 8 in Electric Circuits, 9 th Ed. by Nilsson 1 Chapter 8 EGR 272 – Circuit Theory II 2 nd -order circuits have 2 independent.
Title Subtitle 1. A. B. C. C. * D. Click to try again! INCORRECT.
3 Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.
3.2 Chapter 3 Quadratic Equations. To solve quadratic equations by factoring, apply the which states that, if the product of two real numbers is zero,
EOC Practice #19 SPI EOC Practice #19 Find the solution of a quadratic equation and/or zeros of a quadratic function.
SJTU1 Chapter 7 First-Order Circuit. SJTU2 1.RC and RL Circuits 2.First-order Circuit Complete Response 3.Initial and Final Conditions 4.First-order Circuit.
BALANCING 2 AIM: To solve equations with variables on both sides.
ALGEBRAIC EXPRESSIONS Step 1Write the problem. Step 2Substitute in the values for the unknown (variable). Step 3Use PEMDAS (remember to go left to right).
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